Spectra based endpointing for chemical mechanical polishing

ABSTRACT

Methods and apparatus for spectrum-based endpointing. An endpointing method includes selecting two or more reference spectra. Each reference spectrum is a spectrum of white light reflected from a film of interest on a first substrate and has a thickness greater than a target thickness. The reference spectra is selected for particular spectra-based endpoint determination logic so that the target thickness is achieved when endpoint is called by applying the particular spectra-based endpoint logic. The method includes obtaining two or more current spectra. Each current spectrum is a spectrum of white light reflected from a film of interest on a second substrate when the film of interest is being subjected to a polishing step and has a current thickness that is greater than the target thickness. The method includes determining, for the second substrate, when an endpoint of the polishing step has been achieved.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority to U.S. ProvisionalApplication Ser. No. 60/710,682, filed Aug. 22, 2005. This applicationfurther claims the benefit of priority to and is a continuation-in-partof U.S. application Ser. No. 11/213,344, filed Aug. 26, 2005, entitled“SPECTRUM BASED ENDPOINTING FOR CHEMICAL MECHANICAL POLISHING”.

BACKGROUND

The present invention relates to generally to chemical mechanicalpolishing of substrates.

An integrated circuit is typically formed on a substrate by thesequential deposition of conductive, semiconductive, or insulativelayers on a silicon wafer. One fabrication step involves depositing afiller layer over a non-planar surface and planarizing the filler layer.For certain applications, the filler layer is planarized until the topsurface of a patterned layer is exposed. A conductive filler layer, forexample, can be deposited on a patterned insulative layer to fill thetrenches or holes in the insulative layer. After planarization, theportions of the conductive layer remaining between the raised pattern ofthe insulative layer form vias, plugs, and lines that provide conductivepaths between thin film circuits on the substrate. For otherapplications, such as oxide polishing, the filler layer is planarizeduntil a predetermined thickness is left over the non planar surface. Inaddition, planarization of the substrate surface is usually required forphotolithography.

Chemical mechanical polishing (CMP) is one accepted method ofplanarization. This planarization method typically requires that thesubstrate be mounted on a carrier or polishing head. The exposed surfaceof the substrate is typically placed against a rotating polishing diskpad or belt pad. The polishing pad can be either a standard pad or afixed abrasive pad. A standard pad has a durable roughened surface,whereas a fixed-abrasive pad has abrasive particles held in acontainment media. The carrier head provides a controllable load on thesubstrate to push it against the polishing pad. A polishing slurry istypically supplied to the surface of the polishing pad. The polishingslurry includes at least one chemically reactive agent and, if used witha standard polishing pad, abrasive particles.

One problem in CMP is determining whether the polishing process iscomplete, i.e., whether a substrate layer has been planarized to adesired flatness or thickness, or when a desired amount of material hasbeen removed. Overpolishing (removing too much) of a conductive layer orfilm leads to increased circuit resistance. On the other hand,underpolishing (removing too little) of a conductive layer leads toelectrical shorting. Variations in the initial thickness of thesubstrate layer, the slurry composition, the polishing pad condition,the relative speed between the polishing pad and the substrate, and theload on the substrate can cause variations in the material removal rate.These variations cause variations in the time needed to reach thepolishing endpoint. Therefore, the polishing endpoint cannot bedetermined merely as a function of polishing time.

SUMMARY

In one general aspect, the invention features a computer implementedmethod that includes selecting two or more reference spectra. Eachreference spectrum is a spectrum of white light reflected from a film ofinterest that is on a first substrate and that has a thickness greaterthan a target thickness. The reference spectra are empirically selectedfor particular spectra based endpoint determination logic so that thetarget thickness is achieved when endpoint is called by applying theparticular spectra based endpoint logic. The method includes obtainingtwo or more current spectra. Each current spectrum is a spectrum ofwhite light reflected from a film of interest that is on a secondsubstrate and that has a current thickness that is greater than thetarget thickness. The film of interest on the second substrate issubject to a polishing step. The method includes determining, for thesecond substrate, whether an endpoint of the polishing step has beenachieved, the determining being based on the reference spectra and thecurrent spectra.

In another general aspect, the invention features a computer programproduct that is tangibly stored on machine readable medium. The productincludes instructions operable to cause a processor to select two ormore reference spectra. Each reference spectrum is a spectrum of whitelight reflected from a film of interest that is on a first substrate andthat has a thickness greater than a target thickness. The referencespectra are empirically selected for particular spectra based endpointdetermination logic so that the target thickness is achieved whenendpoint is called by applying the particular spectra based endpointlogic. The product further comprises instructions to obtain two or morecurrent spectra. Each current spectrum is a spectrum of white lightreflected from a film of interest that is on a second substrate and thathas a current thickness that is greater than the target thickness. Thefilm of interest on the second substrate is subject to a polishing step.The product further comprising instructions to determine, for the secondsubstrate, whether an endpoint of the polishing step has been achieved,the determining being based on the reference spectra and the currentspectra.

As used in the instant specification, the term substrate can include,for example, a product substrate (e.g., which includes multiple memoryor processor dies), a test substrate, a bare substrate, and a gatingsubstrate. The substrate can be at various stages of integrated circuitfabrication, e.g., the substrate can be a bare wafer, or it can includeone or more deposited and/or patterned layers. The term substrate caninclude circular disks and rectangular sheets.

Possible advantages of implementations of the invention can include oneor more of the following. Endpoint determination can be made virtuallywithout consideration of variations in polishing rate. Factors thataffect polishing rate, for example, consumables, generally need not beconsidered. The use of multiple reference and/or target spectra (asoppose to a single reference spectrum and/or a single target spectrum)improves accuracy in endpoint determination by providing a difference orendpoint trace that is generally smoother than a trace generated byusing a single reference-spectrum technique. A flushing system can beless likely to dry out slurry on a substrate surface being polished. Apolishing pad window can enhance the accuracy and/or precision ofendpoint determination.

The details of one or more embodiments of the invention are set forth inthe accompanying drawings and the description below. Other features,aspects, and advantages of the invention will become apparent from thedescription, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a chemical mechanical polishing apparatus.

FIGS. 2A-2H show implementations of a polishing pad window.

FIG. 3 shows an implementation of a flushing system.

FIG. 4 shows an alternative implementation of the flushing system.

FIG. 5 is an overhead view of a polishing pad and shows locations wherein-situ measurements are taken.

FIG. 6A shows a spectrum obtained from in-situ measurements.

FIG. 6B illustrates the evolution of spectra obtained from in-situmeasurements as polishing progresses.

FIG. 7A shows a method for obtaining a target spectrum.

FIG. 7B shows a method for obtaining a reference spectrum.

FIGS. 8A and 8B show a method for endpoint determination.

FIGS. 9A and 9B show an alternative method for endpoint determination.

FIGS. 10A and 10B show another alternative method for endpointdetermination.

FIG. 11 shows an implementation for determining an endpoint.

FIG. 12 illustrates peak-to-trough normalization of a spectrum.

FIG. 13 illustrates the smoothing effect using multiple referencespectra provides.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

FIG. 1 shows a polishing apparatus 20 operable to polish a substrate 10.The polishing apparatus 20 includes a rotatable disk-shaped platen 24,on which a polishing pad 30 is situated. The platen is operable torotate about axis 25. For example, a motor can turn a drive shaft 22 torotate the platen 24. The polishing pad 30 can be detachably secured tothe platen 24, for example, by a layer of adhesive. When worn, thepolishing pad 30 can be detached and replaced. The polishing pad 30 canbe a two-layer polishing pad with an outer polishing layer 32 and asofter backing layer 34.

Optical access 36 through the polishing pad is provided by including anaperture (i.e., a hole that runs through the pad) or a solid window. Thesolid window can be secured to the polishing pad, although in someimplementations the solid window can be supported on the platen 24 andproject into an aperture in the polishing pad. The polishing pad 30 isusually placed on the platen 24 so that the aperture or window overliesan optical head 53 situated in a recess 26 of the platen 24. The opticalhead 53 consequently has optical access through the aperture or windowto a substrate being polished. The optical head is further describedbelow.

The window can be, for example, a rigid crystalline or glassy material,e.g., quartz or glass, or a softer plastic material, e.g., silicone,polyurethane or a halogenated polymer (e.g., a fluoropolymer), or acombination of the materials mentioned. The window can be transparent towhite light. If a top surface of the solid window is a rigid crystallineor glassy material, then the top surface should be sufficiently recessedfrom the polishing surface to prevent scratching. If the top surface isnear and may come into contact with the polishing surface, then the topsurface of the window should be a softer plastic material. In someimplementations the solid window is secured in the polishing pad and isa polyurethane window, or a window having a combination of quartz andpolyurethane. The window can have high transmittance, for example,approximately 80% transmittance, for monochromatic light of a particularcolor, for example, blue light or red light. The window can be sealed tothe polishing pad 30 so that liquid does not leak through an interfaceof the window and the polishing pad 30.

In one implementation, the window includes a rigid crystalline or glassymaterial covered with an outer layer of a softer plastic material. Thetop surface of the softer material can be coplanar with the polishingsurface. The bottom surface of the rigid material can be coplanar withor recessed relative to the bottom surface of the polishing pad. Inparticular, if the polishing pad includes two layers, the solid windowcan be integrated into the polishing layer, and the bottom layer canhave an aperture aligned with the solid window.

Assuming that the window includes a combination of a rigid crystallineor glassy material and a softer plastic material, no adhesive need beused to secure the two portions. For example, in one implementation, noadhesive is used to couple the polyurethane portion to the quartzportion of the window. Alternatively, an adhesive that is transparent towhite light can be used or an adhesive can be applied so that lightpassing through the window does not pass through the adhesive. By way ofexample, the adhesive can be applied only to the perimeter of theinterface between the polyurethane and quartz portion. A refractiveindex gel can be applied to a bottom surface of the window.

A bottom surface of the window can optionally include one or morerecesses. A recess can be shaped to accommodate, for example, an end ofan optical fiber cable or an end of an eddy current sensor. The recessallows the end of the optical fiber cable or the end of the eddy currentsensor to be situated at a distance, from a substrate surface beingpolished, that is less than a thickness of the window. With animplementation in which the window includes a rigid crystalline portionor glass like portion and the recess is formed in such a portion bymachining, the recess is polished so as to remove scratches caused bythe machining. Alternatively, a solvent and/or a liquid polymer can beapplied to the surfaces of the recess to remove scratches caused bymachining. The removal of scratches usually caused by machining reducesscattering and can improve the transmittance of light through thewindow.

FIG. 2A-2H show various implementations of the window. As shown in FIG.2A, the window can have two portions, a polyurethane portion 202 and aquartz portion 204. The portions are layers, with the polyurethaneportion 202 situated on top of the quartz portion 204. The window can besituated in the polishing pad so that the top surface 206 of thepolyurethane layer is coplanar with a polishing surface 208 of thepolishing pad.

As shown in FIG. 2B, the polyurethane portion 202 can have a recess inwhich the quartz portion is situated. A bottom surface 210 of the quartzportion is exposed.

As shown in FIG. 2C, the polyurethane portion 202 can includeprojections, for example, projection 212, that project into the quartzportion 204. The projections can act to reduce the likelihood that thepolyurethane portion 202 will be pulled away from the quartz portion 204due to friction from the substrate or retaining ring.

As shown in FIG. 2D, the interface between the polyurethane portion 202and quartz portion 204 can be a rough surface. Such a surface canimprove the strength of the coupling of the two portions of the window,also reducing the likelihood the polyurethane portion 202 will be pulledaway from the quartz portion 204 due to friction from the substrate orretaining ring.

As shown in FIG. 2E, the polyurethane portion 202 can have non-uniformthickness. The thickness at a location that would be in the path 214 ofa light beam is less than the thickness at a location that would not bein the path 214 of the light beam. By way of example, thickness t₁ isless than thickness t₂. Alternatively, the thickness can be less at theedges of the window.

As shown in FIG. 2F, the polyurethane portion 202 can be attached to thequartz portion 204 by use of an adhesive 216. The adhesive can beapplied so that it would not be in the path 214 of the light beam.

As shown in FIG. 2G, the polishing pad can include a polishing layer anda backing layer. The polyurethane portion 202 extends through thepolishing layer and at least partially into the backing layer. The holein the backing layer can be larger in size than the hole in thepolishing layer, and the section of the polyurethane in the backinglayer can be wider than the section of the polyurethane in the polishinglayer. The polishing layer thus provides a lip 218 which overhangs thewindow and which can act to resist a pulling of the polyurethane portion202 away from the quartz portion 204. The polyurethane portion 202conforms to the holes of the layers of the polishing pad.

As shown in FIG. 2H, refractive index gel 220 can be applied to thebottom surface 210 of the quartz portion 204 so as to provide a mediumfor light to travel from a fiber cable 222 to the window. The refractiveindex gel 220 can fill the volume between the fiber cable 222 and thequartz portion 204 and can have a refractive index that matches or isbetween the indices of refraction of the fiber cable 222 and the quartzportion 204.

In implementations where the window includes both quartz andpolyurethane portions, the polyurethane portion should have a thicknessso that, during the life time of the polishing pad, the polyurethaneportion will not be worn so as to expose the quartz portion. The quartzcan be recessed from the bottom surface of the polishing pad, and thefiber cable 222 can extend partially into the polishing pad.

The above described window and polishing pad can be manufactured using avariety of techniques. The polishing pad's backing layer 34 can beattached to its outer polishing layer 32, for example, by adhesive. Theaperture that provides optical access 36 can be formed in the pad 30,e.g., by cutting or by molding the pad 30 to include the aperture, andthe window can be inserted into the aperture and secured to the pad 30,e.g., by an adhesive. Alternatively, a liquid precursor of the windowcan be dispensed into the aperture in the pad 30 and cured to form thewindow. Alternatively, a solid transparent element, e.g., the abovedescribed crystalline or glass like portion, can be positioned in liquidpad material, and the liquid pad material can be cured to form the pad30 around the transparent element. In either of the later two cases, ablock of pad material can be formed, and a layer of polishing pad withthe molded window can be scythed from the block.

With an implementation in which the window includes a crystalline orglass like first portion and a second portion made of soft plasticmaterial, the second portion can be formed in the aperture of the pad 30by applying the described liquid precursor technique. The first portioncan then be inserted. If the first portion is inserted before the liquidprecursor of the second portion is cured, then curing can bond the firstand second portions. If the first portion is inserted after the liquidprecursor is cured, then the first and second potions can be secured byusing an adhesive.

The polishing apparatus 20 can include a flushing system to improvelight transmission through the optical access 36. There are differentimplementations of the flushing system. With implementations of thepolishing apparatus 20 in which the polishing pad 30 includes anaperture instead of a solid window, the flushing system is implementedto provide a laminar flow of a fluid, e.g., a gas or liquid, across atop surface of the optical head 53. (The top surface can be a topsurface of a lens included in the optical head 53.) The laminar flow offluid across the top surface of the optical head 53 can sweep opaqueslurry out of the optical access and/or prevent slurry from drying onthe top surface and, consequently, improves transmission through theoptical access. With implementations in which the polishing pad 30includes a solid window instead of an aperture, the flushing system isimplemented to direct a flow of gas at a bottom surface of the window.The flow of gas can prevent condensation from forming at the solidwindow's bottom surface which would otherwise impede optical access.

FIG. 3 shows an implementation of the laminar-flow flushing system. Theflushing system includes a gas source 302, a delivery line 304, adelivery nozzle 306, a suction nozzle 308, a vacuum line 310, and avacuum source 312. The gas source 302 and vacuum source can beconfigured so that they can introduce and suction a same or a similarvolume of gas. The delivery nozzle 306 is situated so that the laminarflow of gas is directed across the transparent top surface 314 of thein-situ monitoring module and not directed at the substrate surfacebeing polished. Consequently, the laminar flow of gas does not dry outslurry on a substrate surface being polished, which can undesirablyaffect polishing.

FIG. 4 shows an implementation of the flushing system for preventing theformation of condensation on a bottom surface of the solid window. Thesystem reduces or prevents the formation of condensation at the bottomsurface of the polishing pad window. The system includes a gas source402, a delivery line 404, a delivery nozzle 406, a suction nozzle 408, avacuum line 410, and a vacuum source 412. The gas source 402 and vacuumsource can be configured so that they can introduce and suction a sameor a similar volume of gas. The delivery nozzle 406 is situated so thatthe flow of gas is directed at the bottom surface window in thepolishing pad 30.

In one implementation that is an alternative to the implementation ofFIG. 4, the flushing system does not include a vacuum source or line. Inlieu of these components, the flushing system includes a vent formed inthe platen so that the gas introduced into the space underneath thesolid window can be exhausted to a side of the platen or, alternatively,to any other location in the polishing apparatus that can toleratemoisture.

The above described gas source and vacuum source can be located awayfrom the platen so that they do not rotate with the platen. In thiscase, a rotational coupler for convey gas is included each of the supplyline and the vacuum line.

Returning to FIG. 1, the polishing apparatus 20 includes a combinedslurry/rinse arm 39. During polishing, the arm 39 is operable todispense slurry 38 containing a liquid and a pH adjuster. Alternative,the polishing apparatus includes a slurry port operable to dispenseslurry onto polishing pad 30.

The polishing apparatus 20 includes a carrier head 70 operable to holdthe substrate 10 against the polishing pad 30. The carrier head 70 issuspended from a support structure 72, for example, a carousel, and isconnected by a carrier drive shaft 74 to a carrier head rotation motor76 so that the carrier head can rotate about an axis 71. In addition,the carrier head 70 can oscillate laterally in a radial slot formed inthe support structure 72. In operation, the platen is rotated about itscentral axis 25, and the carrier head is rotated about its central axis71 and translated laterally across the top surface of the polishing pad.

The polishing apparatus also includes an optical monitoring system,which can be used to determine a polishing endpoint as discussed below.The optical monitoring system includes a light source 51 and a lightdetector 52. Light passes from the light source 51, through the opticalaccess 36 in the polishing pad 30, impinges and is reflected from thesubstrate 10 back through the optical access 36, and travels to thelight detector 52.

A bifurcated optical cable 54 can be used to transmit the light from thelight source 51 to the optical access 36 and back from the opticalaccess 36 to the light detector 52. The bifurcated optical cable 54 caninclude a “trunk” 55 and two “branches” 56 and 58.

As mentioned above, the platen 24 includes the recess 26, in which theoptical head 53 is situated. The optical head 53 holds one end of thetrunk 55 of the bifurcated fiber cable 54, which is configured to conveylight to and from a substrate surface being polished. The optical head53 can include one or more lenses or a window overlying the end of thebifurcated fiber cable 54 (e.g., as shown in FIG. 3). Alternatively, theoptical head 53 can merely hold the end of the trunk 55 adjacent thesolid window in the polishing pad. The optical head 53 can hold theabove-described nozzles of the flushing system. The optical head 53 canbe removed from the recess 26 as required, for example, to effectpreventive or corrective maintenance.

The platen includes a removable in-situ monitoring module 50. Thein-situ monitoring module 50 can include one or more of the following:the light source 51, the light detector 52, and circuitry for sendingand receiving signals to and from the light source 51 and light detector52. For example, the output of the detector 52 can be a digitalelectronic signal that passes through a rotary coupler, e.g., a slipring, in the drive shaft 22 to the controller for the optical monitoringsystem. Similarly, the light source can be turned on or off in responseto control commands in digital electronic signals that pass from thecontroller through the rotary coupler to the module 50.

The in-situ monitoring module can also hold the respective ends of thebranch portions 56 and 58 of the bifurcated optical fiber 54. The lightsource is operable to transmit light, which is conveyed through thebranch 56 and out the end of the trunk 55 located in the optical head53, and which impinges on a substrate being polished. Light reflectedfrom the substrate is received at the end of the trunk 55 located in theoptical head 53 and conveyed through the branch 58 to the light detector52.

In one implementation, the bifurcated fiber cable 54 is a bundle ofoptical fibers. The bundle includes a first group of optical fibers anda second group of optical fibers. An optical fiber in the first group isconnected to convey light from the light source 51 to a substratesurface being polished. An optical fiber in the second group isconnected to received light reflecting from the substrate surface beingpolished and convey the received light to a light detector. The opticalfibers can be arranged so that the optical fibers in the second groupform an X-like shape that is centered on the longitudinal axis of thebifurcated optical fiber 54 (as viewed in a cross section of thebifurcated fiber cable 54). Alternatively, other arrangements can beimplemented. For example, the optical fibers in the second group canform V-like shapes that are mirror images of each other. A suitablebifurcated optical fiber is available from Verity Instruments, Inc. ofCarrollton, Tex.

There is usually an optimal distance between the polishing pad windowand the end of the trunk 55 of bifurcated fiber cable 54 proximate tothe polishing pad window. The distance can be empirically determined andis affected by, for example, the reflectivity of the window, the shapeof the light beam emitted from the bifurcated fiber cable, and thedistance to the substrate being monitored. In one implementation, thebifurcated fiber cable is situated so that the end proximate to thewindow is as close as possible to the bottom of the window withoutactually touching the window. With this implementation, the polishingapparatus 20 can include a mechanism, e.g., as part of the optical head53, that is operable to adjust the distance between the end of thebifurcated fiber cable 54 and the bottom surface of the polishing padwindow. Alternatively, the proximate end of the bifurcated fiber cableis embedded in the window.

The light source 51 is operable to emit white light. In oneimplementation, the white light emitted includes light havingwavelengths of 200-800 nanometers. A suitable light source is a xenonlamp or a xenon-mercury lamp.

The light detector 52 can be a spectrometer. A spectrometer is basicallyan optical instrument for measuring properties of light, for example,intensity, over a portion of the electromagnetic spectrum. A suitablespectrometer is a grating spectrometer. Typical output for aspectrometer is the intensity of the light as a function of wavelength.

Optionally, the in-situ monitoring module 50 can include other sensorelements. The in-situ monitoring module 50 can include, for example,eddy current sensors, lasers, light emitting diodes, and photodetectors.With implementations in which the in-situ monitoring module 50 includeseddy current sensors, the module 50 is usually situated so that asubstrate being polished is within working range of the eddy currentsensors.

The light source 51 and light detector 52 are connected to a computingdevice operable to control their operation and to receive their signals.The computing device can include a microprocessor situated near thepolishing apparatus, e.g., a personal computer. With respect to control,the computing device can, for example, synchronize activation of thelight source 51 with the rotation of the platen 24. As shown in FIG. 5,the computer can cause the light source 51 to emit a series of flashesstarting just before and ending just after the substrate 10 passes overthe in-situ monitoring module. (Each of points 501-511 depictedrepresents a location where light from the in-situ monitoring moduleimpinged and reflected off.) Alternatively, the computer can cause thelight source 51 to emit light continuously starting just before andending just after the substrate 10 passes over the in-situ monitoringmodule.

With respect to receiving signals, the computing device can receive, forexample, a signal that carries information describing a spectrum of thelight received by the light detector 52. FIG. 6A shows examples of aspectrum measured from light that is emitted from a single flash of thelight source and that is reflected from the substrate. Spectrum 602 ismeasured from light reflected from a product substrate. Spectrum 604 ismeasured from light reflected from a base silicon substrate (which is awafer that has only a silicon layer). Spectrum 606 is from lightreceived by the optical head 53 when there is no substrate situated overthe optical head 53. Under this condition, referred to in the presentspecification as a dark condition, the received light is typicallyambient light.

The computing device can process the above-described signal to determinean endpoint of a polishing step. Without being limited to any particulartheory, the spectra of light reflected from the substrate 10 evolve aspolishing progresses. FIG. 6B provides an example of the evolution aspolishing of a film of interest progresses. The different lines ofspectrum represent different times in the polishing. As can be seen,properties of the spectrum of the reflected light changes as a thicknessof the film changes, and particular spectrums are exhibited byparticular thicknesses of the film. The computing device can executelogic that determines, based on one or more of the spectra, when anendpoint has been reached. The one or more spectra on which an endpointdetermination is based can include a target spectrum, a referencespectrum, or both.

As used in the instant specification, a target spectrum refers to aspectrum exhibited by the white light reflecting from a film of interestwhen the film of interest has a target thickness. By way of example, atarget thickness can be 1, 2, or 3 microns. Alternatively, the targetthickness can be zero, for example, when the film of interest is clearedso that an underlying film is exposed.

There can be and usually are multiple target spectra for a particularthickness of interest. Such is the case because polishing usually occursat a finite rate so that the film of interest maintains the targetthickness for a duration during which multiple spectra can be collected.Moreover, different regions of a patterned substrate usually yielddifferent spectra (even when the spectra were obtained at a same pointof time during polishing). For example, a spectrum of the lightreflecting off a scribe line in a substrate is different (e.g., have adifferent shape) from the spectrum of the light reflecting off an arrayof the substrate. Such a phenomenon is referred to in the instantspecification as a pattern effect. Thus, there can be multiple spectrafor a particular target thickness, and the multiple spectra can includespectra that are different from each other because of pattern effects.

FIG. 7A shows a method 700 for obtaining one or more target spectra.Properties of a substrate with the same pattern as the product substrateare measured (step 702). The substrate which is measured is referred toin the instant specification as a “set-up” substrate. The set-upsubstrate can simply be a substrate which is similar or the same to theproduct substrate, or the set-up substrate could be one substrate from abatch. The properties can include a pre-polished thickness of a film ofinterest at a particular location of interest on the substrate.Typically, the thicknesses at multiple locations are measured. Thelocations are usually selected so that a same type of die feature ismeasured for each location. Measurement can be performed at a metrologystation.

The set-up substrate is polished in accordance with a polishing step ofinterest and spectra of white light reflecting off a substrate surfacebeing polished are collected during polishing (step 704). Polishing andspectra collection can be performed at the above described polishingapparatus. Spectra are collected by the in-situ monitoring system duringpolishing. Multiple spectra can be collected for each platen revolution.The substrate is overpolished, i.e., polished past an estimatedendpoint, so that the spectrum of the light that reflected from thesubstrate when the target thickness is achieved can be obtained.

Properties of the overpolished substrate are measured (step 706). Theproperties include post-polished thicknesses of the film of interest atthe particular location or locations used for the pre-polishmeasurement.

The measured thicknesses and the collected spectra are used to select,from among the collected spectra, one or more spectra determined to beexhibited by the substrate when it had a thickness of interest (step708). In particular, linear interpolation can be performed using themeasured pre-polish film thickness and post-polish substrate thicknessesto determine which of the spectra was exhibited when the target filmthickness was achieved. The spectra determined to be the ones exhibitedwhen the target thickness was achieved are designated to be the targetspectra for the batch of substrates. Typically, three of the collectedspectra are designated to be target spectra. Alternatively, five, seven,and nine spectra are designated to be target spectra.

Optionally, the spectra collected are processed to enhance accuracyand/or precision. The spectra can be processed, for example: tonormalize them to a common reference, to average them, and/or to filternoise from them. Particular implementations of these processingoperations are described below.

As used in the instant specification, a reference spectrum refers to aspectrum that is associated with a target film thickness. Two or morereference spectra are usually empirically selected for particularspectra based endpoint determination logic so that the target thicknessis achieved when the computer device calls endpoint by applying theparticular spectra based endpoint determination logic. The referencespectra can be iteratively selected, as will be described below inreference to FIG. 7B. Reference spectra are usually not the targetspectra. Rather, reference spectra are usually the spectra of lightreflected from the substrate when the film of interest has a thicknessgreater than the target thickness.

FIG. 7B shows a method 701 for selecting two or more reference spectrafor a particular target thickness and particular spectra based endpointdetermination logic. A set up substrate is measured and polished asdescribed above in steps 702-706 (step 703). In particular, spectracollected and the time at which each collected spectrum is measured isstored. Multiple spectra are be collected for each platen revolutionduring the polishing.

A polishing rate of the polishing apparatus for the particular set-upsubstrate is calculated (step 705). The average polishing rate PR can becalculated by using the pre and post-polished thicknesses T1, T2, andthe actual polish time, PT, e.g., PR=(T2−T1)/PT.

An endpoint time is calculated for the particular set-up substrate toprovide a calibration point to test the reference spectrum, as discussedbelow (step 707). The endpoint time can be calculated based on thecalculated polish rate PR, the pre-polish starting thickness of the filmof interest, ST, and the target thickness of the film of interest, TT.The endpoint time can be calculated as a simple linear interpolation,assuming that the polishing rate is constant through the polishingprocess, e.g., ET=(ST−TT)/PR.

Optionally, the calculated endpoint time can be evaluated by polishinganother substrate of the batch of patterned substrates, stoppingpolishing at the calculated endpoint time, and measuring the thicknessof the film of interest. If the thickness is within a satisfactory rangeof the target thickness, then the calculated endpoint time issatisfactory. Otherwise, the calculated endpoint time can bere-calculated.

Two or more of the collected spectra are selected and designated to bethe reference spectra (step 709). The spectra selected to be thereference spectra are the spectra of light reflected from the substratewhen the film of interest has a thickness greater than and approximatelyequal to the target thickness. Typically, three of the collected spectraare designated to be reference spectra. Alternatively, five, seven, ornine spectra are designated to be reference spectra. As with the targetspectra, there can be multiple reference spectra because the polishingrate is finite.

In one implementation, the particular platen revolution corresponding tothe endpoint time calculated in step 707 is identified, and spectracollected during the particular platen revolution are selected to bedesignated as reference spectra. By way of example, spectra collectedcan be from a center region of the substrate. The platen revolutioncorresponding to the endpoint time calculated is the platen revolutionduring which a time corresponding to the calculated endpoint timeoccurred. By way of example, if the calculated endpoint time was 25.5seconds, then the particular platen revolution corresponding to thiscalculated endpoint time is the platen revolution during which 25.5second of polishing has occurred in the polishing process.

The use of multiple reference spectra

The particular spectra based endpoint determination logic is executed insimulation using the spectra collected for the set-up substrate and withthe selected two or more spectra designated to be the reference spectra(step 711). Execution of the logic yields an empirically derived butsimulated endpoint time that the logic has determined to be theendpoint.

The empirically derived but simulated endpoint time is compared to thecalculated endpoint time (step 713). If the empirically derived endpointtime is within a threshold range of the calculated endpoint time, thenthe currently selected two or more reference spectra are known togenerate a result that matches the calibration point. Thus, when theendpoint logic is executed using the reference spectra in a run-timeenvironment, the system should reliably detect an endpoint at the targetthickness. Therefore, the spectra currently designated to be thereference spectra can be kept as the reference spectra for run timepolishing of the other substrates of the batch (step 715). Otherwise,steps 709 and 711 are repeated as appropriate.

Optionally, variables other than the selected spectra can be changed foreach iteration (i.e., each performance of steps 709 and 711). Forexample, the above-mentioned processing of the spectra (for example,filter parameters) and/or a threshold range from a minimum of adifference trace can be changed. The difference trace and the thresholdrange of a minimum of the difference trace are described below.

FIG. 8A shows a method 800 for using spectra based endpointdetermination logic to determine an endpoint of a polishing step.Another substrate of the batch of patterned substrates is polished usingthe above-described polishing apparatus (step 802). At each revolutionof the platen, the following steps are performed.

Two or more spectra of white light reflecting off a substrate surfacebeing polished are measured, and two or more current spectra for acurrent platen revolution are obtained (step 804). Spectra measure atpoints 501-511 (FIG. 5) are examples of spectra measured during thecurrent platen revolution. The spectra measured during the currentplaten revolution are optionally processed to enhance accuracy and/orprecision as described above in reference to FIG. 7A and as describedbelow in reference to FIG. 11.

Two or more of the spectra measured during the current platen revolutionare selected to be the current spectra for the current platenrevolution. In one implementation, the spectra selected to be currentspectra are those measured at sample locations near the center of thesubstrate (for example, at points 505, 506, and 507 shown in FIG. 5).The selected spectra are not averaged, and each selected spectra isdesignated to be a current spectrum for the current platen revolution.

A difference between each of the current spectra and each of thereference spectra is calculated (step 806). The reference spectra can beobtained as described above in reference to FIG. 7B. In oneimplementation, the difference is a sum of differences in intensitiesover a range of wavelengths. That is,

${Difference} = {\sum\limits_{\lambda = a}^{b}{{abs}\left( {{I_{current}(\lambda)} - {I_{reference}(\lambda)}} \right)}}$where a and b are the lower limit and upper limit of the range ofwavelengths of a spectrum, respectively, and I_(current)(λ) andI_(reference)(λ) are the intensity of a current spectra and theintensity of the target spectra for a given wavelength, respectively.

One way to calculate a difference between each of the current spectraand each of the reference spectra is to select each of the currentspectra. For each selected current spectra, the difference is calculatedagainst each of the reference spectra. Given current spectra e, f and g,and reference spectra E, F, and G, for example, a difference would becalculated for each of the following combinations of current andreference spectra: e and E, e and F, e and G, f and E, f and F, f and G,g and E, g and F, and g and G.

The smallest of the calculated differences is appended to a differencetrace (step 808). The difference trace is usually updated once perplaten revolution. The difference trace is generally a plot of one ofthe calculated differences (in this case the smallest of the differencescalculated for the current platen revolution). As an alternative to thesmallest difference, another of the differences, for example, a medianof the differences or the next to smallest difference, can be appendedto the trace.

Taking the smallest of the differences can improve accuracy in theendpoint determination process. The current spectra can include spectrathat are from light reflecting off different locations (e.g., a scribeline and an array) of the substrate, and the above-described patterneffect can cause these spectra to be significantly different. Likewise,the reference spectra can include spectra that are from light reflectingoff different locations on the substrate. A comparison of such differentspectra are faulty and can introduce error into endpoint determination.For example, a comparison of a current spectrum of light reflecting offa scribe line of a patterned substrate to a reference spectrum of lightreflecting off an array of the patterned substrate can introduce errorinto the endpoint determination calculus. Such a comparison,figuratively speaking, is a comparison between apples and oranges. Inconsidering only the smallest of the differences, these types ofcomparisons (even though they are made) are factored out of thecalculus. Thus, by using multiple reference spectra and multiple currentspectra, and by considering only the smallest of the differences betweeneach of these spectra, errors that would be introduced by the describedfaulty comparison are avoided.

Optionally, the difference trace can be processed, for example,smoothing the difference trace by filtering out a calculated differencethat deviates beyond a threshold from preceding one or more calculateddifferences.

Whether the difference trace is within a threshold value of a minimum isdetermined (step 810). After the minimum has been detected, the endpointis called when the different trace begins to rise past a particularthreshold value of the minimum. Alternatively, the endpoint can becalled based on the slope of the difference trace. In particular, theslope of the difference trace approaches and becomes zero at the minimumof the difference trace. The endpoint can be called when the slope ofthe difference trace is within a threshold range of the slope that isnear zero.

Optionally, window logic can be applied to facilitate the determinationof step 808. Window logic suitable for use is described in commonlyassigned U.S. Pat. Nos. 5,893,796 and 6,296,548, which are incorporatedby reference.

If the difference trace is NOT determined to have reached a thresholdrange of a minimum, polishing is allowed to continue and steps 804, 806,808, and 810 are repeated as appropriate. Otherwise, an endpoint iscalled and polishing is stopped (step 812).

FIG. 8B illustrates the above described method for determining endpoint.Trace 801 is the raw difference trace. Trace 803 is the smootheddifference trace. Endpoint is called when the smoothed difference trace803 reaches a threshold value 805 above the minimum 807.

As an alternative to using reference spectra, target spectra can be usedin the method 800. The difference calculation would be between currentspectra and target spectra, and endpoint would be determined when thedifference trace reaches a minimum.

FIG. 9A shows an alterative method 900 for using a spectrum-basedendpoint determination logic to determine an endpoint of a polishingstep. A set-up substrate is polished and target spectra and referencespectra are obtained (step 902). These spectra can be obtained asdescribed above in reference to FIGS. 7A and 7B.

A target difference is calculated (step 904). The target difference isthe minimum of the differences between the reference spectra and thetarget spectra, which are calculated using the above-describeddifference equation and method for calculating differences (i.e., step808).

Polishing of another substrate of the batch of substrates is started(step 906). The following steps are performed for each platen revolutionduring polishing. Two or more spectra of white light reflecting off asubstrate surface being polished are measured to obtain two or morecurrent spectra for a current platen revolution (step 908). Differencesbetween the current spectra and the reference spectra are calculated(step 910). The smallest of the calculated differences is appended to adifference trace (step 912). (Steps 908, 910, and 912 are similar tosteps 804, 806, and 808, respectively.) Whether the difference trace iswithin a threshold range of the target difference is determined (step914). If the difference trace is NOT determined to have reached athreshold range of the target difference, polishing is allowed tocontinue and steps 908, 910, 912, and 914 are repeated as appropriate.Otherwise, an endpoint is called and polishing is stopped (step 916).

FIG. 9B illustrates the above described method for determining endpoint.Trace 901 is the raw difference trace. Trace 903 is the smootheddifference trace. Endpoint is called when the smooth difference trace903 is within a threshold range 905 of a target difference 907.

FIG. 10A shows another method 1000 for determining an endpoint of apolishing step. Reference spectra are obtained (step 1002). Thereference spectra are obtained as described above in reference to FIG.7B.

The spectra collected from the process of obtaining the referencespectrum are stored in a library (step 1004). Alternatively, the librarycan include spectra that are not collected but theoretically generated.The spectra stored, including the reference spectra, are indexed so thateach spectrum has a unique index value. The indexing is implemented sothat the index values are sequenced in an order in which the spectrawere measured. The index, thus, can be correlated to time and/or platenrevolution. In one implementation, a first spectrum collected at a firstpoint in time will have an index value that is less than a secondspectrum that is collected at a later point in time. The library can beimplemented in memory of the computing device of the polishingapparatus.

A substrate from the batch of substrates is polished, and the followingsteps are performed for each platen revolution. Two or more spectra aremeasured to obtain two or more current spectra for a current platenrevolution (step 1006). The spectra are obtained as described above.Each current spectra is compared to the spectra stored in the library,and the library spectrum which best fits any of the current spectra isdetermined (step 1008). The index of the library spectrum determined tobest fit any of the current spectra is appended to an endpoint indextrace (step 1010). Endpoint is called when the endpoint trace reaches anindex of any of the reference spectra (step 1012).

FIG. 10B illustrates the above described method for determiningendpoint. Trace 1014 is the raw index trace. Trace 1016 is the smootheddifference trace. Line 1018 represents the index value of the referencespectrum. Multiple current spectra can be obtained in each sweep of theoptical head beneath the substrate, e.g., a spectra for each radial zoneon the substrate being tracked, and an index trace can be generated foreach radial zone.

FIG. 11 shows an implementation for determining an endpoint during apolishing step. For each platen revolution, the following steps areperformed. Multiple raw spectra of white light reflecting off asubstrate surface being polished are measured (step 1102).

Each measured raw spectra is normalized to remove light reflectionscontributed by mediums other than the film or films of interest (step1104). Normalization of spectra facilitates their comparison to eachother. Light reflections contributed by media other than the film orfilms of interest include light reflections from the polishing padwindow and from the base silicon layer of the substrate. Contributionsfrom the window can be estimated by measuring the spectrum of lightreceived by the in-situ monitoring system under a dark condition (i.e.,when no substrates are placed over the in-situ monitoring system).Contributions from the silicon layer can be estimated by measuring thespectrum of light reflecting of a bare silicon substrate. Thecontributions are usually obtained prior to commencement of thepolishing step.

A measured raw spectrum is normalized as follows:normalized spectrum=(A−Dark)/(Si−Dark)where A is the raw spectrum, Dark is the spectrum obtained under thedark condition, and Si is the spectrum obtained from the bare siliconsubstrate.

Optionally, the collected spectra can be sorted based on the region ofthe pattern that has generated the spectrum, and spectra from someregions can be excluded from the endpoint calculation. In particular,spectra that are from light reflecting off scribe lines can be removedfrom consideration (step 1106). As discussed above, different regions ofa pattern substrate usually yield different spectra (even when thespectra were obtained at a same point of time during polishing). Forexample, a spectrum of the light reflecting off a scribe line in asubstrate is different from the spectrum of the light reflecting off anarray of the substrate. Because of their different shapes, use ofspectra from both regions of the pattern usually introduces error intothe endpoint determination. However, the spectra can be sorted based ontheir shapes into a group for scribe lines and a group for arrays.Because there is often greater variation in the spectra for scribelines, usually these spectra can be excluded from consideration toenhance precision.

Step 1106 can be an alternative to the technique of using multiplereference spectra (described above in step 808 of method 800) tocompensate for the above-described errors caused by faulty comparisons.Step 1106 can be performed in lieu of or in addition to step 808.

A subset of the spectra processed thus far is selected (step 1108). Thesubset consists of the spectra obtained from light reflecting off thesubstrate at points of a region on the substrate. The region can be, forexample, region 512 (FIG. 5).

Optionally, a high-pass filter is applied to the measured raw spectra(step 1110). Application of the high pass filter typically removes lowfrequency distortion of the average of the subset of spectra. Thehigh-pass filter can be applied to the raw spectra, their average, or toboth the raw spectra and their average.

Each spectrum of the subset of spectra is normalized so that itsamplitude is the same or similar to the amplitude of a referencespectrum (step 1112). The amplitude of a spectrum is the peak-to-troughvalue of the spectrum. Alternatively, each spectrum of the subset ofspectra is normalized so that its amplitude is the same or similar to areference amplitude to which the reference spectra have also beennormalized.

A difference between each of the normalized spectrum and each referencespectrum is calculated (step 1114). The reference spectra are obtainedas described in reference to FIG. 7B. The difference is calculated usingthe above-described equation for calculating differences betweenspectra.

A difference trace is updated with the smallest of the calculateddifferences (step 1116). The difference trace exhibits a calculateddifferences between normalized spectra and reference spectra as afunction of time (or platen revolution).

A median filter and a low-pass filter are applied to the updateddifference trace (step 1118). Application of these filters typicallysmoothes the trace (by reducing or eliminating spikes in the trace).

Endpoint determination is performed based on the updated and filtereddifference trace (step 1120). The determination is made based on whenthe difference trace reaches a minimum. The above described window logicis used to make the determination.

More generally, the signal processing steps of steps 1104-1112 can beused to improve endpoint determination procedures. For example, insteadof generation of a difference trace, the normalized spectra could beused to select a spectra from a library to generate an index trace, asdescribed above in reference to FIG. 10A.

FIG. 12 illustrates the normalization of step 1112. As can be seen, onlya portion of a spectrum (or an average of spectra) is considered fornormalization. The portion considered is referred to in the instantspecification as a normalization range and, furthermore, can be userselectable. Normalization is effected so that the highest point and thelowest point in the normalization range are normalized to 1 and 0,respectively. The normalization is calculated as follows:g=(1−0)/(r _(max) −r _(min))h=1−r _(max) ·gN=R·g+h

where, g is a gain, h is an offset, r_(max) is the highest value in thenormalization range, r_(min) is the lowest value in the normalizationrange, N is the normalized spectrum, and R is the pre normalizedspectrum.

FIG. 13 illustrates the smoothing effect using multiple referencespectra provides. Trace 1302 was generated using a single referencespectrum (which is an average). Trace 1304 was generated using threereference spectra (as described above in reference to FIG. 8). Trace1306 was generated using nine reference spectra. As can be seen, trace1304 includes fewer spikes than does trace 1302. That is, the trace 1304is smoother than trace 1302. Moreover, trace 1304 has a more defined dipthan does trace 1306, which is significant because it is the dip thatallows endpoint determination logic to call endpoint. A more defined dipthus facilitates endpoint determination.

Embodiments of the invention and all of the functional operationsdescribed in this specification can be implemented in digital electroniccircuitry, or in computer software, firmware, or hardware, including thestructural means disclosed in this specification and structuralequivalents thereof, or in combinations of them. Embodiments of theinvention can be implemented as one or more computer program products,i.e., one or more computer programs tangibly embodied in an informationcarrier, e.g., in a machine-readable storage device or in a propagatedsignal, for execution by, or to control the operation of, dataprocessing apparatus, e.g., a programmable processor, a computer, ormultiple processors or computers. A computer program (also known as aprogram, software, software application, or code) can be written in anyform of programming language, including compiled or interpretedlanguages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a computing environment. A computer program does notnecessarily correspond to a file. A program can be stored in a portionof a file that holds other programs or data, in a single file dedicatedto the program in question, or in multiple coordinated files (e.g.,files that store one or more modules, sub-programs, or portions ofcode). A computer program can be deployed to be executed on one computeror on multiple computers at one site or distributed across multiplesites and interconnected by a communication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application-specific integrated circuit).

The above described polishing apparatus and methods can be applied in avariety of polishing systems. Either the polishing pad, or the carrierhead, or both can move to provide relative motion between the polishingsurface and the substrate. For example, the platen may orbit rather thanrotate. The polishing pad can be a circular (or some other shape) padsecured to the platen. Some aspects of the endpoint detection system maybe applicable to linear polishing systems, e.g., where the polishing padis a continuous or a reel-to-reel belt that moves linearly. Thepolishing layer can be a standard (for example, polyurethane with orwithout fillers) polishing material, a soft material, or afixed-abrasive material. Terms of relative positioning are used; itshould be understood that the polishing surface and substrate can beheld in a vertical orientation or some other orientation.

Particular embodiments of the invention have been described. Otherembodiments are within the scope of the following claims. For example,the actions recited in the claims can be performed in a different orderand still achieve desirable results.

1. A computer program product, tangibly stored on machine readablemedium, the product comprising instructions operable to cause aprocessor to: select two or more reference spectra, each referencespectrum representing a spectrum of light reflected from a film ofinterest on a first substrate when illuminated with white light; obtaintwo or more current spectra, each current spectrum being a spectrum oflight reflected from a film of interest on a second substrate whenilluminated with white light, the film of interest on the secondsubstrate being subject to a polishing step, and the two or more currentspectra being obtained from a single platen rotation of the polishingstep; calculate differences between the current spectra and thereference spectra to generate a plurality of differences for the singleplaten rotation; select a smallest difference from the plurality ofdifferences; and determine, for the second substrate, whether anendpoint of the polishing step has been achieved, the determining beingbased on the smallest difference.
 2. The computer program product ofclaim 1, further comprising instructions to call the endpoint when thesmallest difference reaches a threshold value.
 3. The computer programproduct of claim 2, wherein: instructions to obtain current spectraincludes instructions to obtain current spectra for each revolution of aplaten used in the polishing step; and instructions to determine whetherthe endpoint of the polishing step has been achieved includesinstructions to generate a difference trace that includes multiplepoints, each of which represents the smallest of the differencescalculated for a revolution of the platen.
 4. The computer programproduct of claim 3, wherein: instructions to determine whether theendpoint of the polishing step has been achieved includes instructionsto determine whether the difference trace has reached a minimum.
 5. Thecomputer program product of claim 4, wherein: instructions to determinewhether the endpoint of the polishing step has been achieved includesinstructions to determine whether the difference trace has risen to athreshold value above the minimum.
 6. The computer program product ofclaim 4, wherein: instructions to determine whether the difference tracehas reached a minimum includes instructions to calculate a slope of thedifference trace.
 7. The computer program product of claim 3, furthercomprising instructions to: apply a filter to smooth the trace.
 8. Thecomputer program product of claim 1, wherein each current spectrum ofthe current spectra and each reference spectrum of the reference spectraexhibits a peak-to-trough amplitude and the product further comprisesinstructions to: normalize the current spectra so that thepeak-to-trough amplitude of each current spectrum is substantially thesame as the peak-to-trough amplitude of one of the reference spectrum.9. The computer program product of claim 1, further comprisinginstructions to: normalize each of the current spectra and the referencespectra to remove light reflections contributed by mediums other thanthe films of interest.
 10. The computer program product of claim 1,wherein: instruct:ions to obtain current spectra include instructions toobtain a current spectrum that is not from light that reflected off ascribe line of the second substrate.
 11. The computer program product ofclaim 1, wherein: the reference spectra is empirically selected.
 12. Thecomputer program product of claim 1, wherein the reference spectra aretheoretically generated.
 13. A computer program product, tangibly storedon machine readable medium, the product comprising instructions operableto cause a processor to: select two or more reference spectra, eachreference spectrum representing a spectrum of light reflected from afilm of interest that is on a first substrate and that has a thicknessgreater than a target thickness, the reference spectra being selectedfor particular spectra based endpoint determination logic so that thetarget thickness is achieved when endpoint is called by applying theparticular spectra based endpoint logic; obtain two or more currentspectra, each current spectrum being a spectrum of light reflected froma film of interest that is on a second substrate, is subject to apolishing step and has a current thickness that is greater than thetarget thickness, wherein the instructions to obtain current spectrainclude instructions to obtain current spectra for each revolution of aplaten used in the polishing step; determine, for the second substrate,whether an endpoint of the polishing step has been achieved, thedetermining being based on the reference spectra and the currentspectra, including calculating a difference between each of thereference spectra and each of the current spectra and generating adifference trace that includes multiple points, each of which representsthe smallest of the differences calculated for a revolution of theplaten; and call the endpoint when one of the differences reaches athreshold value.
 14. The computer program product of claim 13, wherein:instructions to determine whether the endpoint of the polishing step hasbeen achieved includes instructions to determine whether the differencetrace has reached a minimum.
 15. The computer program product of claim14, wherein: instructions to determine whether the endpoint of thepolishing step has been achieved includes instructions to determinewhether the difference trace has risen to a threshold value above theminimum.
 16. The computer program product of claim 14, wherein:instructions to determine whether the difference trace has reached aminimum includes instructions to calculate a slope of the differencetrace.
 17. The computer program product of claim 13, further comprisinginstructions to: apply a filter to smooth the trace.